Case A and B evolution towards electron capture supernova
aa r X i v : . [ a s t r o - ph . S R ] J u l Astronomy & Astrophysics manuscript no. 32502 © ESO 2018July 12, 2018
Case A and B evolution towards electron capture supernova
L. Siess , and U. Lebreuilly Institut d’Astronomie et d’Astrophysique, Universit´e Libre de Bruxelles (ULB), CP 226, B-1050 Brussels, Belgium Max-Planck-Institut f¨ur Astrophysik, Karl Schwarzschild Str. 1, 85741 Garching, GermanyReceived 20 December 2017/ Accepted 3 March 2018
ABSTRACT
Context.
Most super AGB stars are expected to end their life as oxygen-neon white dwarfs rather than electron capturesupernovae (ECSN). The reason is ascribed to the ability of the second dredge-up to significantly reduce the mass ofthe He core and of the efficient AGB winds to remove the stellar envelope before the degenerate core reaches the criticalmass for the activation of electron capture reactions.
Aims.
In this study, we investigate the formation of ECSN through case A and case B mass transfer. In these scenarios,when Roche lobe overflow stops, the primary has become a helium star. With a small envelope left, the second dredge-upis prevented, potentially opening new paths to ECSN.
Methods.
We compute binary models using our stellar evolution code
BINSTAR . We consider three different secondarymasses of 8, 9, and 10 M ⊙ and explore the parameter space, varying the companion mass, orbital period, and inputphysics. Results.
Assuming conservative mass transfer, with our choice of secondary masses all case A systems enter contact eitherduring the main sequence or as a consequence of reversed mass transfer when the secondary overtakes its companionduring core helium burning. Case B systems are able to produce ECSN progenitors in a relatively small range of periods(3 < ∼ P (d) ≤
30) and primary masses (10 . ≤ M/M ⊙ ≤ . M /M remains less than 1 . − .
5, above which evolution to contact becomesunavoidable. We also find that allowing for systemic mass loss substantially increases the period interval over whichECSN can occur. This change in the binary physics does not however affect the primary mass range. We finally stressthat the formation of ECSN progenitors through case A and B mass transfer is very sensitive to adopted binary andstellar physics.
Conclusions.
Close binaries provide additional channels for ECSN but the parameter space is rather constrained likelymaking ECSN a rare event.
Key words. stars:binaries – white dwarfs – supernovae:general
1. Introduction
In the past decade, super-AGB (SAGB) stars have gener-ated a resurgence of interest in the stellar evolution com-munity; they represent a non-negligible fraction of starsin the Galaxy and until recently their contribution to thegalactic chemical enrichment was largely ignored, mainlydue to the lack of reliable yields. Their peculiar evolutionand in particular their fate as electron capture supernovae(ECSN) is also a matter of active research. The final evo-lution of SAGB stars still represents a challenge to one-dimensional (1D) modelling with off-center neon ignitionand/or silicon burning flame that propagate to the center(Woosley & Heger 2015), possibly leading in some cases tothe disruption of the SAGB core and the formation of ironremnants (Jones et al. 2016). Hydrodynamical simulationsof ECSN (for a review, see M¨uller 2016) provide success-ful explosions after core bounce and subsequent neutrinoheating. These supernovae produce the lowest mass neutronstars and because the explosion is fast and not very pow-erful ( ≈ erg), the neutron star is expected to receive asmall natal velocity kick. This weak impulse is invoked toexplain the neutron star retention problem in globular clus-ters (Pfahl et al. 2002a) or the small eccentricity of some Send offprint requests to : L. Siess high-mass X-ray binaries such as X Per (Pfahl et al. 2002b).We note however that there is no clear evidence for a bi-modal velocity distribution in pulsars kicks (Hobbs et al.2005) that would identify the neutron star explosion mech-anism as EC or core collapse (CC) supernova.SAGB stars are members of a specific class of starssqueezed between intermediate-mass stars that end theirlives as CO white dwarfs (WD) and massive stars that formneutron stars or black holes after they explode as core col-lapse supernova (CCSN). The mass of SAGB stars rangesbetween M up the minimum mass for carbon ignition and M mas the minimum mass for CCSN. The value of thesedividing masses critically depends on the adopted mixingprescription at the edge of the convective core and variesbetween M up ≈ − M ⊙ and M mas ≈ − M ⊙ (e.g.,Siess 2006). The evolution of SAGB stars (for a recent re-view, see Doherty et al. 2017) is characterized by the off-center ignition of carbon followed by the propagation ofa deflagration front to the center and the formation of anoxygen-neon core. The subsequent evolution depends on theability of the degenerate core to reach the critical mass of M EC = 1 . M ⊙ (Nomoto 1984) above which the densityis high enough for electron capture reactions on the abun-dant Ne to take place. If this threshold is reached, thereduction of the electron number induces the collapse of
1. Siess: Case A and B evolution towards electron capture supernova the stellar core and the formation of a low-mass neutronstar. On the other hand, if the SAGB star is able to getrid of its envelope because of efficient wind mass loss forinstance, the end product of the evolution is an ONe WD.The critical mass that delineates these two fates is usuallyreferred to as M n , the minimum mass for the formation ofa neutron star.Results of full stellar evolution calculations indicate thatthe mass range of single stars that undergo an ECSN is verynarrow, with M mas − M n ≈ . − . M ⊙ (Doherty et al.2015). There are two main reasons why so few SAGB starsfollow this explosive path. The first one has to do with theoccurrence of the second dredge-up (2DUP). Indeed, at theend of core helium burning, the expansion of the star to redgiant dimensions is accompanied with the deepening of itsconvective envelope. In SAGB and massive intermediate-mass stars, the surface convection zone reaches the outerHe-rich layers and reduces the He core mass below theChandrasekhar limit (see e.g. Fig 5 of Siess 2006). It isworth reiterating that massive stars do not experience a2DUP event and maintain a massive He core that can sub-sequently evolve through all the nuclear burning stages un-til the formation of an iron core. Second, the mass lossrate during the thermally pulsating SAGB phase is sostrong compared to the core growth rate that the entireSAGB envelope is lost before the core mass can reach M EC .Therefore only stars that enter the SAGB phase after the2DUP with a core mass close to M EC may eventually goSN, the large majority ending as ONe WD. The appar-ent failure of single stars to evolve toward ECSN shouldhowever be mitigated because a large fraction of stars arein binary systems (Raghavan et al. 2010; Duchˆene & Kraus2013) and will interact with their companion at some pointduring their evolution (Sana et al. 2012). Among these in-teracting systems, some ECSN progenitors may emerge.In this paper, we discuss these binary channels and morespecifically those resulting from the evolution through caseA and case B mass transfer. The following section setsthe stage and reviews the binary paths leading to ECSN(Sect. 2). After a description of our code and physical as-sumptions (Sect. 3), we present the results of our simula-tions, starting with a description of the evolution of repre-sentative case A and case B systems. We then explore the ef-fect of varying the initial period, the mass ratio and the as-sumptions concerning conservative mass transfer (Sect. 4).In Sect. 5, we analyze the mass of the He star after caseB Roche lobe overflow (RLOF) and of the envelope at thepre-ECSN stage. In Sect. 6, we discuss the uncertaintiesaffecting this modeling and compare our results with therecent work of Poelarends et al. (2017) before concludingin Sect 7.
2. The binary channels to ECSN
Single SAGB stars fail to explode as ECSN because theirONe core mass cannot reach the critical value of 1.37 M ⊙ .One way to overcome this problem is to increase the coremass by adding matter on top of it. Accretion from a binarycompanion is an obvious means to do this, but depending onthe composition of the accreted material, different scenarioshave to be considered: either a CV-like evolution in the case We define the mass ratio as the ratio of the actual donor’smass to that of the gainer: q = M /M . MS + MSunstable RLOF M > M giant + MS Fig. 1.
Cartoon illustrating the formation of an EC progen-itor in a single (left) and double degenerate (right) scenario.of accretion of H-rich material (Sect. 2.1) or the merger oftwo WDs when the accreted matter is made of C and O(Sect. 2.2). In Sect. 2.3 we describe the evolution of heliumstars whose characteristics are similar to those of the barecore of SAGB stars except that their envelope has beenremoved.
Short period binaries consisting of an O+Ne WD accret-ing material from a main sequence companion representthe high-mass analogs of cataclysmic variables (CV) andthe formation of these systems is similar to that of CVs.In this paradigm (Ritter 2012), a SAGB primary fills itsRoche lobe as it ascends its thermally pulsing SAGB phase(Fig. 1, left). Because the donor star has an extended con-vective envelope, it expands upon mass loss and mass trans-fer becomes dynamically unstable . In this process, thelow-mass companion cannot assimilate the matter that israpidly dumped on its surface and the binary becomes im-mersed in a common envelope. During the subsequent evo-lution ( ∼ This requires however that at the start of Roche lobe over-flow the mass ratio is large enough ( q = M /M > ∼ . − .
5, seeWebbink 1988)2. Siess: Case A and B evolution towards electron capture supernova has been attested by observations of neon novae (e.g.,Starrfield et al. 1986; Downen et al. 2013). The fate of theWD is then mainly dictated by the mass accretion rate(Nomoto 1982; Nomoto et al. 2007): if the mass transfer ishigher than the core growth rate then the H-rich materialcannot be assimilated by the WD and expands to red-giantdimensions, potentially leading to a CE evolution. On theother hand, if it is too low, the H-shell becomes unsta-ble and recurrent novae-like flashes develop that removemost of the accreted mass, thus preventing core growth. Inthe intermediate regime, steady H-burning takes place andthe core mass can increase. When the WD reaches M EC ,EC reactions start on Mg and when the central density ρ c ≈ × g cm − they proceed on the abundant Ne(Nomoto 1984, 1987). These reactions induce a rapid con-traction of the core, which is accelerated by the strong de-pendence of the EC rates on density, and the rise of thetemperature by the emission of γ -rays. When the centraltemperature T c ≈ × K, oxygen burning starts, but theoutcome of the evolution (collapse to a neutron star or coredisruption) depends on the density ρ ig at which oxygen isignited. If the density is high enough, ρ ig > ∼ × g cm − ,EC on the NSE material left behind by the passage of theoxygen deflagration is fast enough to induce gravitationalcollapse before the expansion induced by thermonuclear en-ergy release quenches them (Isern et al. 1991; Canal et al.1992; Gutierrez et al. 1996).As reported by various authors, many uncertainties inthe adopted physics impact ρ ig and thus the outcome (e.g.,Isern & Hernanz 1994). Among them is the critical treat-ment of mixing (Gutierrez et al. 1996). In the early stageof collapse, semi-convection develops at the center. Usingthe Ledoux instead of the Schwarzschild criterion wouldthen lead to a less efficient cooling of the central regionand thus to a higher central temperature and lower den-sity at the time of oxygen ignition. Isern et al. (1991) alsoshowed that if the velocity of the deflagration front, whichgoverns the nuclear energy production, exceeds some crit-ical value (which depends on the mode of energy trans-port, conduction or convection), then EC reactions on theNSE material behind the oxygen deflagration front maynot be fast enough to induce collapse. A recent study bySchwab et al. (2015) confirms the absence of convection atthe time of activation of the EC reactions implying low igni-tion densities ( ρ ig > ∼ . × g cm − ) but the authors alsoconclude that at this density, the star should still collapseto a neutron star. Using the model of Schwab et al. (2015)as initial conditions for their 3D hydrodynamical simula-tions, Jones et al. (2016) find on the contrary that the coreis partially disrupted. Explosive oxygen burning provokesthe ejection of a fraction of the WD material and leadsto the formation of a bound remnant composed of O-Neand Fe-group elements (see also Isern et al. 1991). On theother hand, if a neutron star is formed, the result of thisAIC scenario may be the formation of a millisecond pulsar(Tauris et al. 2013b). The formation of a double degenerate system is illus-trated in Fig. 1 (right) and involves two common envelopephases. The merger of two WD was first modeled using1D stellar evolution codes by Nomoto & Iben (1985) and Saio & Nomoto (1985). In this scenario, examined earlierby Iben & Tutukov (1984) and Webbink (1984), the systemcomes into contact due to the loss of angular momentum bygravitational wave emission. The least massive WD, whichhas the larger radius, overfills its Roche lobe and is subse-quently tidally disrupted. The matter is then assumed todistribute in a disk allowing the deposition of CO rich ma-terial on the more massive WD. The authors showed thatif the mass accretion rate is high enough (higher than 1/5of the Eddington limit, ∼ × − M ⊙ yr − ), then carbonignites off-center and is followed by the propagation of aburning front that converts the entire CO core in an ONemixture, like in SAGB stars. If the mass of the merger ex-ceeds M EC , then an ECSN is a likely outcome. On the otherend, if the mass accretion rate is too small, the inner shellsheat faster than the surface layers and the central ignition ofcarbon at higher density leads to a SNIa explosion. Duringthe merger process, the WD may be significantly spun upbut the effects of stellar rotation have been shown to besmall (Saio & Nomoto 2004).This classical picture has recently been contested inlight of hydrodynamical calculations of WD merger events.These simulations (e.g., Shen et al. 2012; Raskin et al.2012) show that once the lower-mass WD fills its Rochelobe, mass transfer becomes unstable and leads to the com-plete disruption of the star. At the end of this dynamicalphase that lasts ∼ − s, the accretor is surroundedby a fast rotating hot envelope attached to a centrifugallysupported thick disk. A stream of matter is also present inthe simulations but the material remains bound to the sys-tem and eventually falls back onto the accretor. Shen et al.(2012) simulations also indicate that after the merger prod-uct has reached a quasi-hydrostatic equilibrium configura-tion, magneto-rotational instabilities develop in the dis-rupted WD material and efficiently redistribute the an-gular momentum. According to their simulations, within10 − s, the remnant evolves towards an equilibrium con-figuration of shear-free solid body rotation. In this process,viscous heating has substantially raised the temperatureof the envelope, potentially allowing for C ignition alreadyduring the merger phase. During this “viscous-phase”, themerger product reaches a nearly spherical geometry, witha cool CO WD at the center, surrounded by a thermallysupported envelope (Schwab et al. 2012). These conditionsare quite different from those used in earlier studies wherethe primary was accreting mass at a nearly Eddington ratefrom a keplerian disk. In this new configuration, there is noaccretion and the evolution of the remnant is determinedby the cooling and heat redistribution in the hot envelope.Yoon et al. (2007) were the first to implement the resultsof a 3D SPH simulation of a WD merger into a 1D stellarevolution code. However, they did not consider the effect ofMHD instabilities on the redistribution of angular momen-tum, so their starting structure is slightly different, con-sisting of a cool WD surrounded by a hot envelope gainingmass from an accretion disk. In their analysis, they derivethe conditions for off-center carbon ignition and show thatthey depend on the initial temperature, mass accretion rate,and efficiency of angular momentum transport. Using morerealistic initial conditions that take into account the vis-cous phase, Schwab et al. (2016) followed the evolution of a0.9+0.6 M ⊙ merger. In their model carbon ignites off-centerduring the dynamical phase. The propagation of a carbonburning front towards the center is similar to that occurring
3. Siess: Case A and B evolution towards electron capture supernova
Fig. 2.
Cartoon illustrating the possible occurrence ofECSN after the formation of a He star in the scenariosinvolved for the formation of high- (left) and low- (right)mass X-ray binaries. This cartoon has been adapted fromTauris & van den Heuvel (2006). The numbers indicate themasses of the stellar components.in SAGB stars and results in the formation of a ONe core.However, when the carbon flame reaches the center, thedegeneracy has been lifted. The core then contracts, neonignites off-center and a Ne-O deflagration propagates to thecenter converting the core into a hot mixture of Si-groupelements. The final outcome depends on the core mass: ifit is less than ∼ . M ⊙ , the remnant is a silicon WD,otherwise silicon burning proceeds and the core collapsesto a neutron star. The authors suggest that the same con-clusion could have been reached by Nomoto & Iben (1985)if they had been able to run their simulations for a longertime. These results depend, however, on the efficiency ofmass loss during the heat redistribution phase but the im-portant point is that CO WD mergers may not provide aviable path toward ECSN.For the merger of a ONe and CO WDs, Kawai et al.(1987) showed that the compressional heating induced bymass accretion at a nearly Eddington rate is not strongenough to ignite neon off-center. In this case, once a suffi-cient amount of carbon has been accreted, carbon burningoperates above the O+Ne core and contributes to increasethe WD mass. The process of carbon accretion/burning re-peats until the WD mass reaches M EC and EC reactionsbegin. In the early 1980’s, Nomoto (1984, 1987) and Habets (1986)studied the evolution of He stars and showed that mod-els with initial masses ranging between 2.0 M ⊙ and 2.5 M ⊙ would evolve toward ECSN. But these single stellar modelsdo not take into account the effects of binary interactionsthat can deeply impact the progenitor’s structure. Indeed,the final He core mass depends on the presence of a convec-tive envelope because of the contribution coming from theashes of H-shell burning. The models of Wellstein & Langer(1999) show that a donor star that loses its envelope willdevelop a smaller He core compared to a single evolution. Inaddition, the exchange of mass and angular momentum be-tween the binary components can induce hydrodynamicalinstabilities responsible for the transport of chemicals andangular momentum. Rotational mixing can strongly affectthe size of the stellar core (e.g., Maeder & Meynet 2000)and the entire structure up to the point that, if the systemis near contact, the stars may be tidally locked and followa chemically homogeneous evolution (e.g., de Mink et al.2009).Podsiadlowski et al. (2004) also pointed out that if thestar is able to lose its H-rich envelope by the end ofcore He-burning, the second dredge-up can be avoided,thereby removing one of the main factors preventing theevolution toward ECSN. Using Nomoto (1984) He coremass range and binary models from Wellstein et al. (2001),Podsiadlowski et al. (2004) estimated that stars in a binarysystem with masses in the range 8 − M ⊙ would likely un-dergo an ECSN. As we will show, consistent binary modelsconsiderably reduce this primary mass range. The evolu-tion of naked stellar cores has been the subject of variousinvestigations, largely related to the formation of low- andhigh-mass X-ray binaries as well as millisecond pulsars (seee.g., the review by Tauris & van den Heuvel 2006). Heliumstars can form via common envelope evolution or as a re-sult of case A and case B mass transfer and are thus a verycommon outcome of binary evolution.In a recent work, Tauris et al. (2015) analyzed the evo-lution of close binary systems composed of a He star donorand a neutron star companion with initial periods between0.06 and 2 days. This period range was chosen so the pri-mary would fill its Roche lobe during its evolution. Theirsimulations indicate that ECSNs occur over a narrow massrange of ≈ . − . M ⊙ and involve He stars with massesbetween 2.6 M ⊙ and 2.95 M ⊙ (see their Fig. 18).
3. Modeling and Methodology
The calculations presented in this paper have been per-formed with the
BINSTAR code whose detailed descriptioncan be found in Siess et al. (2013) and Davis et al. (2013).In brief, the code solves the structure of the two stars andthe evolution of the orbital parameters (eccentricity andseparation) simultaneously. When the star fills its Rochelobe, mass transfer rate is calculated according to the pre-scriptions of Ritter (1988) and Kolb & Ritter (1990), up-dated at each iteration during the convergence process. TheRoche radius is approximated by the Eggleton (1983) for-mula. In our simulations we consider circular orbits and ne-glect the stellar spins. The exchange of mass between the
4. Siess: Case A and B evolution towards electron capture supernova
Fig. 3.
HR diagram of a representative 11.2+9 M ⊙ , two-day case A evolution. The color bar on the side indicatesthe magnitude of the mass transfer rate.stellar components is accompanied by the transfer of an-gular momentum. If the evolution is not conservative, weassume that the material leaving the system carries awaythe specific angular momentum of the gainer star at its po-sition along the orbit (so-called re-emission mode). In allour models, we use the Asplund et al. (2005) solar compo-sition, neglect stellar winds, and use a mixing length pa-rameter α = 1 .
75. We apply a moderate core overshootingmodeled with the use of an exponentially decaying diffusioncoefficient outside the Schwarzschild boundary with a pa-rameter f over = 0 .
01 (for details of this implementation in
STAREVOL , see Siess 2007). Our nuclear network includesall the necessary reactions to accurately follow the energet-ics up to neon ignition and our simulations are stopped oncethe maximum temperature reaches ∼ . × K. With ourassumptions, we find that single stars in the mass range9 . M ⊙ < ∼ M zams < ∼ . M ⊙ go ECSN. With our limited nuclear network, we are not able to followthe evolution up to the final stage. Therefore, we had to de-vise criteria to determine the most likely fate of our stellarmodels. Our method relies on several indicators. The firstsystematic one is borrowed from Tauris et al. (2015) anddeclares a model as ECSN progenitor if, during the post-carbon burning phase, the central temperature T c does notexceed the maximum value reached during the core carbon-burning phase. According to these authors this correspondsto stars with an ONe core mass in the range 1.37 M ⊙ to1.43 M ⊙ . Therefore any star that ends up with a CO coremass in that range and fulfills the temperature condition isdeclared a ECSN progenitor. The second indicator is mostlya check based on a visual inspection of the evolution ofthe star in the central density versus central temperature Fig. 4.
Evolutionary properties of a representative11.2+9 M ⊙ , two-day case A system. From top to bottom,the panels show the overfilling factor (ratio of the stellar toRoche radius), the Kippenhahn diagrams of the secondaryand of the primary, and the Roche lobe overflow mass trans-fer rate. The red dotted lines in the top and bottom panelsrefer to the secondary.( ρ c − T c ) diagram. We simply make sure that for the selectedECSN progenitor, the increase in the central temperatureafter core carbon burning is modest. Stars that evolve to-wards an ONe WD show a pronounced decrease in theircentral temperature while in massive stars the temperaturerises steadily with increasing density.
4. Results
Before exploring the parameter space leading to ECSN, westart our study with the analysis of two representative sys-tems illustrating a conservative evolution during case A andcase B evolution.
Our prototype is a 11.2 M ⊙ primary with a 9 M ⊙ compan-ion in an initial period of 2 days. The evolution of the sys-tem in the HR diagram is presented in Fig. 3. The primarystarts filling its Roche lobe while on the main sequence.The mass transfer (Fig. 4) shows the typical pattern with arapid, thermally unstable phase during which most of themass is transferred followed by a slower phase where themass transfer is driven by the nuclear expansion of the staron the main sequence. The system temporarily detacheswith the exhaustion of nuclear fuel in the core but masstransfer resumes soon after the establishment of H-shellburning, an episode referred to as case AB. The systemdetaches when the primary contracts as a result of centralhelium ignition. At that stage, the secondary has gained ≈ M ⊙ and the binary is now composed of a 1.4 M ⊙ He
5. Siess: Case A and B evolution towards electron capture supernova
Fig. 5.
HR diagram of a representative 11.2+9 M ⊙ , four-day case B evolution. The color bar on the side indicatesthe magnitude of the mass transfer rate.star primary and a 18.8 M ⊙ main sequence companion or-biting each other with a period of ≈
110 d. We also note inthe Kippenhahn diagram of Fig. 4 the rejuvenation (e.g.,De Greve & Packet 1990) of the secondary characterized bythe increase of its convective core upon accretion. Becauseof its high mass, the secondary now evolves much fasterthan the primary and is able to overtake its companion dur-ing core helium burning. The secondary leaves the He mainsequence first and the subsequent expansion of its radiusleads to a new episode of mass transfer. However given theextreme mass ratio ( M /M >
13) of this system, a com-mon envelope (CE) evolution is unavoidable. This reversecase C evolution is found in all our conservative case Aevolution (see 4.3) and corresponds to the late overtaking type described in Nelson & Eggleton (2001). To guess theoutcome of this system, we followed Dewi & Tauris (2000)and computed the expected final separation assuming thatthe Roche overfilling secondary would lose all its H-rich en-velope, technically defined as the mass coordinate wherethe H mass fraction drops below 0.1. Using a CE efficiencyparameter η CE = 1 and the binding energy computed withour stellar models, we find that at the end of the CE evolu-tion the secondary still fills its Roche lobe ( R L /R ≈ . η CE λ = 1, we find that R L /R ≈ .
8, so in this casea merger is avoided. It is therefore difficult to conclude onthe fate of the binary. We extended our analysis to otherreverse case C systems and reached the same conclusion be-cause all these binaries share the same properties (periodsbetween 80 and 120 d, M around 18 M ⊙ , M of the orderof ∼ . − . M ⊙ and radius R ≈ . − . R ⊙ ).We conclude that with our choice of secondary masses,case A systems fail to produce ECSN progenitors. A suc-cessful evolutionary path requires a more massive primary that can develop a bigger He core and evolve rapidly enoughto avoid being overtaken by its companion (for details seePoelarends et al. 2017). To illustrate case B evolution, we consider the same sys-tem but with a longer initial period of 4 days. The evo-lution in the HR diagram is presented in Fig. 5. Withlarger initial separation, mass transfer begins after the pri-mary has left the main sequence and proceeds on its ther-mal timescale. The donor star being more evolved than incase A, its Kelvin Helmholtz timescale is shorter leadingto a higher mass transfer rate with a maximum value of ∼ × − M ⊙ yr − compared to ∼ − M ⊙ yr − in case A(Fig. 6). In this scenario, the slow phase is absent and whenthe system detaches (after ∼ × yr), the primary, now aHe star, is substantially more massive than in case A witha mass of 2.5 M ⊙ and the period is shorter with P ≈
50 d.The He star has a small H-envelope of < . M ⊙ and ter-minates central He-burning while its companion is still onthe main sequence. The exhaustion of fuel causes the ex-pansion of the primary and triggers a new episode of masstransfer, referred to as case BB. The mass transfer rate isslightly less intense than before ( ˙ M < × − M ⊙ yr − )and stops after ∼ M ⊙ . At the end of the simu-lation, the period is ∼
180 yr and the companion is now amain sequence O star of 18.6 M ⊙ . After the SN explosionof the He star, the system will likely appear as a high massX-ray binary where the neutron star accretes matter fromthe companion’s wind. To analyze how the fate of the primary depends on the ini-tial period, we consider a system with a 9 M ⊙ companion.In the following section, we investigate the effect of chang-ing the mass of the secondary.The results of our binary simulations are presented inFig. 8. Systems with periods shorter than ≈ . § P < ∼ M ⊙ ,1 d system), the secondary is initially close to filling itsRoche lobe and contact occurs during the slow nucleartimescale mass transfer (case AS in Nelson & Eggleton(2001) terminology). With increasing separation (e.g., a11+9 M ⊙ , 1.5 d system) the companion is deeper insideits potential well and accepts the accreted matter with-out overfilling its Roche lobe. However, its evolution is ac-
6. Siess: Case A and B evolution towards electron capture supernova
Fig. 6.
Representative evolution of a 11.2+9 M ⊙ , four-daycase B system. From top to bottom: overfilling factor (ratioof the stellar to Roche radius), Kippenhahn diagrams of thesecondary and of the primary and Roche lobe overflow masstransfer rate. The red dotted lines in the top panel refer tothe secondary.celerated to the extent that it leaves the MS before thedonor star. This situation corresponds to the early over-taking or premature contact described by Wellstein et al.(2001). These authors also emphasized that the occurrenceof contacts depends on the criterion used to define con-vection. In particular, using the Ledoux criterion tends tosuppress rejuvenation (Braun & Langer 1995), resulting insecondaries with smaller cores that have a shorter mainsequence lifetimes and are consequently more prone to pre-mature contact. Pols (1994) analyzed the evolution of closebinaries with donor stars in the mass range 8 − M ⊙ andfound that case A systems with a mass ratio q > ∼ . − . < ∼ P (d) < ∼
20, we see in Fig. 8that a variety of outcomes become possible. Let us analyzefor instance systems with an initial period P = 4 d. Fora donor star with M < . M ⊙ , the He-star left at theend of RLOF has a mass less than ≈ . M ⊙ (see Sect. 5).In these models, carbon ignites off-center and leads to theformation of an ONe core less massive than M EC . The pri-mary thus ends up as a degenerate ONe WD. Between11 . < ∼ M /M ⊙ ≤ .
5, the ONe core fulfills the criteriadefined in Sect. 3.2 and the star evolves towards ECSN.Above M > . M ⊙ , at the end of core carbon burning,the ONe core exceed 1.43 M ⊙ , Ne ignites and the evolutionproceeds to CCSN. It is also expected that if the mass ratiobecomes even larger ( q > . Fig. 7.
As in Fig. 6 but zooming into case BB. Time iscounted backward from the last computed model. Masstransfer stops around log(t − t end ) ≈ . τ KH ) and may have devel-oped a deep convective envelope. Since the maximum mass Fig. 8.
Fate of the primary star as a function of its massand initial period for a system with a 9 M ⊙ secondary com-panion. The triangles indicate an evolution to contact withthe cyan triangles corresponding to systems that undergoa reverse case C mass transfer. The open magenta circlesand black filled squares label binaries in which the primaryend its life as an ONe WD and ECSN, respectively. Thegreen stars identify systems with a massive He core thatevolves as CCSN. The transition between case A and caseB mass transfer is delineated by the dotted line and occursfor periods between 2.3 and 2.5 days.
7. Siess: Case A and B evolution towards electron capture supernova transfer rate scales as ˙ M ≈ M /τ KH , these late case Bsystems will come into contact and enter a CE evolution.This transition to contact occurs with initial periods around P ≈ −
25 d. We reiterated the same procedure describedat the end of Sect. 4.1 to estimate the final separation af-ter the CE evolution. At the time of contact, our late caseB systems typically consist of a H-shell burning donor of5-6 M ⊙ with a He core of 1 . − . M ⊙ and a main se-quence companion of 14 − M ⊙ . With periods in the range P ≈ −
70 d, assuming that the gainer is unaffected inthe process, we find that after the removal of its envelope,the core of the primary still largely overfills its Roche lobe.These systems are thus likely going to merge.Finally, for very long initial periods, the system re-mains detached and one recovers the single star chan-nel for ECSN, i.e., primaries with masses in the range9 . < ∼ M /M ⊙ < ∼ . M ⊙ star is ≈ R ⊙ , binary sys-tems with companion masses of 9 M ⊙ will remain detachedif the initial period P > ∼ The effect of varying the companion’s mass is illustrated inFig. 9. Overall, the dependence on the secondary mass isweak because once RLOF starts, nothing can really preventthe primary’s envelope from being lost. For a given initialseparation, increasing the companion mass will trigger masstransfer earlier because the primary’s Roche radius, whichis an increasing function of the mass ratio, is smaller. As aconsequence the primary mass range for ECSN is slightlyshifted downward when M is larger.For companion masses below M < ∼ M ⊙ , the ECSNchannel is closed because the minimum primary mass re-quired for an ECSN event ( M > ∼ M ⊙ ) yields a mass ratio q > .
5, which is high enough to bring the system into con-tact either during case A or case B evolution. With ourassumptions, we did not find any ECSN progenitors witha 7 M ⊙ secondary and all the systems with q < . In the above simulations, we assumed that mass trans-fer was conservative but a fraction of the mass lost bythe donor may escape from the system. This nonconser-vatism is often advocated in case A and B mass transferto explain for example the mass ratio distribution of Algols(Deschamps et al. 2013). To assess the impact of systemicmass loss, we use a very simple model where we assumethat a constant fraction β RLOF = / of the transferredmass leaves the system, carrying away the specific orbital Fig. 9.
As in Fig. 8 but for a 8 M ⊙ (right panel) and 10 M ⊙ (left panel) secondary.angular momentum of the gainer star. In this so-called re-emission mode, the torque applied on the orbit writes˙ J Σ = β RLOF ˙ M a Ω , (1)where ˙ M < a the dis-tance between the gainer and the center of mass, and Ω theorbital angular velocity. Interestingly, with this systemicangular momentum loss prescription, the separation startsto increase as soon as the mass ratio q < ∼ . , which isearlier than in the standard case where this occurs when q < β − longer than in a standard evolution. Asillustrated in Fig. 10, the expansion of the gainer is no-ticeably slower and much smaller when mass is allowed toescape from the system. So when the secondary reachesits maximum radius, the period has substantially increasedand the star is unable to fill its Roche lobe. In our ex-ample (a 10.9+9 M ⊙ , 70 d binary), the Roche filling fac-tor R /R L < . P > ∼
100 d go into contact. For comparison, this thresholdperiod is of the order of 20 d in the conservative case. Thedecrease in the ECSN progenitor mass with increasing pe-riod is also present in these liberal models and since thechannel is now open to longer-period systems, less massiveprimaries can go SN but the width in progenitor’s massesstill remains quite narrow.
5. He star masses
The bottom panel of Fig. 12 shows the mass of the pri-mary at the end of case B RLOF. All He star primariesare initially surrounded by a H envelope of a few 0.1 M ⊙ but most of this H layer will be converted into He or lostduring the subsequent case BB mass transfer episode. TheHe star mass range of ECSN progenitors is confined to a It is easy to show that under the above assumptions, for β RLOF = / the rate of change of the separation writes˙ a over a = | ˙ M | over q ( M + M ) (cid:0) q over2 − q (cid:1) , (2)and is positive for q < (1 + √ / ≈ . .
8. Siess: Case A and B evolution towards electron capture supernova
Fig. 10.
Variation with time of selected quantities dur-ing the conservative (black, solid line) and nonconservative(red, dotted line) evolution of a 10.9+9 M ⊙ , 70 d systems.From top to bottom are shown, the period in days, the stel-lar masses and the filling factor, mass accretion rate, andradius of the gainer. Fig. 11.
As in Fig. 8 but for a nonconservative evolutionwhere we assume that a constant fraction β RLOF = / ofthe mass lost by the primary escapes from the system.narrow region, between 2.55 M ⊙ and 2.7 M ⊙ . This resultis very similar to that of Tauris et al. (2015) but differentfrom the early models of Nomoto (1984) where it was con-sidered that He stars in the mass range 2 . − . M ⊙ wouldundergo an ECSN. We also see a trend of increasing He starmass with increasing orbital period. This effect is due to theoccurrence of case BB mass transfer. With longer initial pe-riods, the He star fills its Roche lobe later which implies ahigher mass transfer rate and a stronger reduction of itsmass before the explosion. Our simulations indicate that Fig. 12.
Mass of the He star primary ( M He ) at the endof RLOF (bottom), and masses of the CO core (mid-dle) and of the remaining envelope (top) at the endof the simulation in case B mass transfer. The fi-nal stellar mass is the sum of M CO and M env . Thehatched strip in the mid panel indicate the coremass range that we considered for the formationof ECSN progenitors. The color coding is the same asFig. 8.He stars less massive than M He < ∼ . M ⊙ end up as ONewhite dwarfs and that above M He > ∼ . M ⊙ they evolveinto a CCSN. Figure 12 also shows the CO core and enve-lope masses at the end of our simulations. We see that ourpotential ECSN candidates are surrounded by a He layer of0.1 to 0.9 M ⊙ . With increasing period, the He star progeni-tor is more massive and the remaining envelope mass is alsolarger. In this case B scenario, most of our progenitors areexpected to end as SNIb since SNIc are generally attributedto progenitors with thin He envelopes ( M env < ∼ . M ⊙ ,Hachinger et al. 2012) although the classification also de-pends on the amount of mixed Ni in the surface layers. Weshould also stress that the exact envelope mass at the timeof explosion is uncertain because in some cases mass trans-fer is still active and wind mass loss, which has not beenconsidered in these simulations, may be strong in these lu-minous He stars. In terms of CO core masses, the massinterval is slightly smaller and better defined with ECSNprogenitors having 2 . < ∼ M CO /M ⊙ < ∼ .
6. Discussion
The determination of the primary mass range of ECSN pro-genitors is unfortunately affected by many uncertainties.The first one is associated with the treatment of core over-shooting and is independent of the evolutionary scenarios.As shown in various studies (e.g., Siess 2007; Gil-Pons et al.2007; Farmer et al. 2015), the implementation of extra-mixing at the edge of the convective core has a dramatic
9. Siess: Case A and B evolution towards electron capture supernova impact on the values of M up and M mas with variations upto 2 − . M ⊙ . Calculations using different levels of over-shooting or different algorithms to define the convectiveboundaries will therefore introduce some inevitable scatterin the derived progenitors mass range.Another issue is related to the difficulty of assessing thefinal evolution of some models. In some simulations, wefind that at the end of carbon burning, neon ignites off-center. Similarly to off-center carbon ignition, a few neon-oxygen (NeO) shell flashes may develop before a NeO flameforms and propagates to the center. The fate of the starthen depends on whether or not this flame is able to reachthe center before the density reaches the critical thresh-old for electron capture reactions. Jones et al. (2013, 2014)showed that the results depend on the (unknown) mixingprocesses operating at the base of the convective flame. Ifmixing is present, the NeO flame quenches, allowing thecore to contract more efficiently and reach higher densi-ties. If instead the strict Schwarzschild criterion is used,provided neon is not ignited too far off-center, the flamecan reach the center before the URCA reactions start andthe star ends its life as a CCSN. Therefore the fate ofthese models is dictated by the mixing across the burn-ing front which, in the absence of dedicated hydrodynami-cal simulations will remain a serious limitation. Jones et al.(2014) claim however that even a very small amount ofmixing (of the order of 10 − − − times the pressurescale height at the convective front) could disrupt the flameand accelerate the contraction, in which case stars that ig-nite Ne off-center would never undergo an ECSN. Thesetheoretical uncertainties have consequences on the crite-ria used to determine the fate of the star (Sect. 3.2). Wedid a test reducing the maximum ONe core mass of theECSN progenitor from 1.43 M ⊙ down to 1.41 M ⊙ and foundvery small differences in the derived fate. The only af-fected models are those located at the boundary betweenECSN and CCSN which introduces a typical uncertaintyof < ∼ . M ⊙ in the estimated mass range. As outlined inthe previous section, the assumptions concerning mass andangular momentum loss from the system have a deep im-pact on the period range over which ECSNs occur. Variousstudies have shown that mass transfer in binaries is notnecessarily conservative and there is no reason to assumethat our ECSN binary progenitors should evolve as such.This is attested in Algols (e.g., Deschamps et al. 2013;Mennekens & Vanbeveren 2017) as well as in some mas-sive binaries (e.g., de Mink et al. 2007; Mahy et al. 2011).Several scenarios have been devised to account for liberality,including mass-loss enhancement due to the spin-up of theaccretor (e.g., Petrovic et al. 2005; Yoon et al. 2010) andthe release of accretion energy in the hot spot region thatforms where the accretion stream from the primary impactsthe surface of the gainer (van Rensbergen et al. 2008). Inour systems with periods P ≤
20 d, mass transfer occurs viadirect impact and within this latter scenario the accretionefficiency could be significantly reduced. Unfortunately thisprocess is badly understood and the period range of ECSNwill be subject to a large degree of uncertainty dependingon the prescription used to remove mass and angular mo-mentum from the system.We should also bear in mind that among the systemsthat go into contact, in particular during case A mass trans-fer (the reverse case C systems indicated by the cyan tri-angles in Fig. 8, 9, 11), the subsequent common envelope evolution may not necessarily lead to a merger but mayproduce a He star that eventually goes ECSN. Such sce-narios are advocated for the formation of X-ray binaries(Fig. 2) and these systems should be accounted for whenestimating the total number of ECSN binary progenitors.Recently, Poelarends et al. (2017) undertook a very sim-ilar study and a comparison of their work is instructive.Since all our case A systems come into contact (Sect. 4.1),we focus our comparison on case B systems and on theECSN progenitors. A relevant difference between the twoworks in terms of input physics is their use of the Ledouxcriterion to delineate the convective boundary and of semi-convective mixing. These assumptions will produce signif-icantly smaller core compared to our models that con-sider overshooting beyond the Schwarszchild boundary.Comparing their Fig. 12 with our Figs. 8, 9 and 11 indi-cates a systematic shift in the primary mass of ≈ . M ⊙ ,theirs being more massive. Such differences are expected(e.g., Siess 2007) and do not affect the conclusion sharedby the two studies that the formation of ECSN progenitorsrequires a minimum mass ratio of M /M > . − . β = 0 . vs. [3,100]). The differences is likely due tothe higher mass of their systems which, according to Eq. 2induces a smaller rate of expansion of the orbit for a given q and ˙ M . The presence of stellar winds can also contributeto reduce the orbital angular momentum. The other maindifference concerns the primary mass range for ECSN pro-genitors which is ≈ M ⊙ wide for their case B systemscompared to ≈ . M ⊙ in our study. The reason is at-tributed to a very efficient cooling of their structures al-lowing CO cores as massive as 1.52 M ⊙ to go ECSN. Thisis substantially larger than the value we use (1.43 M ⊙ ) andthat was found by Tauris et al. (2015). It should be notedthat using a pre-ECSN CO core mass of 1.43 M ⊙ in theirFig. 12 considerably reduces the progenitor mass range andreconciles the two studies. In their analysis, they ascribedthe efficient cooling of their models to the intense massloss that affects the ECSN candidate in the last stage ofevolution. Our models also show these high mass trans-fer episodes during C-shell burning (Fig. 7) but they donot produce the amount of cooling the MESA models ex-perience. Tauris et al. (2015) did not report this featureeither. At this stage it is difficult to understand this differ-ent behavior, which may be related to a much higher massloss rate in their simulations (they report values as high as˙
M > − M ⊙ yr − ), to the equation of state, to our lim-ited network, or to numerics (we discard the neutrino lossrate as the same prescriptions are used in both codes).To conclude, the two studies are in good qualitative agree-ment but differences in the cooling efficiencies of the ONecore in the last stage of the evolution has a significant effecton the final mass range of ECSN progenitors.
10. Siess: Case A and B evolution towards electron capture supernova
7. Conclusion
As stated above, the period and mass intervals for ECSNprogenitors strongly depend on the treatment of binaryinteractions and stellar physics. Assuming a conservativeevolution, we find that for our choice of secondary massesall case A systems enter a contact phase either when theprimary is on the main sequence or when the gainer over-takes the evolution of the donor when it leaves the corehelium burning phase. On the other hand, we find thatcase B systems with periods 3 < ∼ P (d) < ∼
20 and primarymasses between 10 . ≤ M /M ⊙ ≤ . q < ∼ . ∼
100 d with our as-sumptions) and less massive primaries to go ECSN. Atthe end of RLOF, the mass of the He star progenitorssits in the range 2 . < ∼ M He /M ⊙ < ∼ .
7. In an early study,Podsiadlowski et al. (2004) claimed that stars in a binarysystem with masses between 8 M ⊙ and 11 M ⊙ would likelyundergo an ECSN. Our consistent binary evolution calcu-lations lead to a significant downward revision of this massrange and this conclusion is also shared by Poelarends et al.(2017). Given the strong constraints on the parameters forstars to go ECSN, we are tempted to conclude that theseexplosions are rare, even including binaries. However, pop-ulation studies should be performed to quantify the likeli-hood of these channels and investigate how the probabilitiesdepend on the various uncertainties and scenarios. Acknowledgments
L.S. thanks the Max-Planck Institute for Astrophysics inGarching and in particular Achim Weiss and Ewald M¨ullerfor their hospitality during the final elaboration of thiswork. LS is senior FRS-F.N.R.S. research associate.
References